What is the mathematical relationship between the velocity of P waves (v_p) and S waves (v_s)?

The relationship between the velocities of P waves (primary waves) and S waves (secondary waves) in a homogeneous medium is derived from the properties of the materials through which these waves are traveling. P waves are compressional waves that travel through both solids and liquids, while S waves are shear waves that can only travel through solids.

In general, the velocity of P waves is greater than that of S waves due to the differences in the way these waves propagate through materials. The mathematical relationship is given by the formula v_p = sqrt(3) * v_s, which reflects the fact that the velocity of P waves is dependent on the density and elastic properties of the material.

The factor of sqrt(3) arises from the physics of wave propagation, illustrating that P waves traverse through the material with a higher velocity than S waves because they utilize the compressional properties of the medium more effectively. This relationship is fundamental in geophysics and helps seismologists interpret seismic data to understand Earth's internal structure.